mm-1008 === Subject: Re: Please help. Having problem with simple math logic >I have 2 number scales. >Top scale = 42 through -84 >and >Bottom scale = 17 through 782 >If 17 on the bottom scale represents 42 on the top scale, and the same >goes for 782 on the bottom with -84 on the top, how do I Žgure out >any number on the top scale by inputting a number on the bottom scale. >(i.e. 261 on the bottom scale = ? on the top scale. You mean e.g., not i.e. (Why do people use words when they don¹t know the meaning?) But there¹s a more substantive issue: You haven¹t told us what relationship exists between the two scales. Is it supposed to be a straight-line relationship, one where e.g. any interval of 100 points on the bottom scale makes the same width interval on the top scale? If you make that assumption, your problem can be solved. You have two (x,y) points, namely (17, 42) and (782, -48), and you need to Žnd the equation of the line between them. The slope of that line is -90/765, so you use the point-slope form: top = (-90/765)*(bottom-17) + 782 or top = (-2/17)*bottom + 784 -- Stan Brown, Oak Road Systems, Cortland County, New York, USA http://OakRoadSystems.com Address munging may or may not reduce the spam you get; it surely reduces the number of useful answers you get. http://www.cs.tut.Ž/~jkorpela/usenet/laws.html === Subject: Re: Please help. Having problem with simple math logic >>Top scale = 42 through -84 >>Bottom scale = 17 through 782 >If you make that assumption, your problem can be solved. You have >two (x,y) points, namely (17, 42) and (782, -48), and you need to >Žnd the equation of the line between them. The slope of that line >is -90/765, so you use the point-slope form: > top = (-90/765)*(bottom-17) + 782 Well _That_ was careless of me! The given point is (17,42), so the equation should be top = (-90/765)(*bottom-17) + 42 or top = (-2/17)*bottom + 44 which also works with the point (782, -48) as it should. -- Stan Brown, Oak Road Systems, Cortland County, New York, USA http://OakRoadSystems.com Address munging may or may not reduce the spam you get; it surely reduces the number of useful answers you get. http://www.cs.tut.Ž/~jkorpela/usenet/laws.html === Subject: Re: ReL Please help. Having problem with simple logic. >>Top scale = 42 through -84 >>Bottom scale = 17 through 782 > The formula is T= (-0.1657)B+ 170.8. That a is negative is due to >the fact that the two scales are reversed, a higher value on one >corresponds to a lower value on the other. The given point (17,42) does not work in that formula. -- Stan Brown, Oak Road Systems, Cortland County, New York, USA http://OakRoadSystems.com Address munging may or may not reduce the spam you get; it surely reduces the number of useful answers you get. http://www.cs.tut.Ž/~jkorpela/usenet/laws.html === Subject: Re: Solving normal probability >I need some serious help with probability. I just cannot get it. And what have you done to try to get it, other than simply copy the problem into your news posting software? Please tell us what you tried, and _speciŽcally_ where you got stuck. If you need a hint, here¹s one: Your second and third questions both require you to apply the Empirical Rule. -- Stan Brown, Oak Road Systems, Cortland County, New York, USA http://OakRoadSystems.com Address munging may or may not reduce the spam you get; it surely reduces the number of useful answers you get. http://www.cs.tut.Ž/~jkorpela/usenet/laws.html === Subject: Re: Yes, your Želd is corrupt > Some of you may have noticed that I¹ve Žnally focused on the base > assumption required to Žght the proof that there is an over one > hundred year old deŽnition in core mathematics, as posters are > Žnally revealed to be arguing for a relation like xy=uf, where x and > y are algebraic integer functions. > It¹s like someone arguing with you about xy=2, being in the ring of > integers, where x can be any integer, as I can even set u=1, f=2, to > *get* that value. > So how have they gotten away with arguing such a position for months? > You and other mathematicians have let them because mathematics is just > a joke to you, and maybe a way to make money. > Maybe you know that people just *trust* mathematicians, and you¹re in > the Želd to be in an area where the truth doesn¹t matter. > Your Želd is corrupt. Mathematicians don¹t really care about > anything but themselves. They are sick and lost. > You are lost if you are with them. > James Harris > This is a crock of bull. If you are right, then I think everyone who has > taken math needs to go back and relearn the correct mathematics. You need to > grow up; don¹t worry about this fame and money you claim that you deserve. > You have a physics degree; USE IT! > What would you do if someone attacked the Želd of physics? It¹d be the same > thing you¹re doing with mathematics. Get help......fast! Oh yeah, try to shift the focus from your *proven* corruption. It¹s mathematicians who are defying mathematics. It¹s mathematicians who are now caught in a stupid lie to hide an over one hundred year old error. It¹s mathematicians who are running from the truth. Your are corrupt and lost souls. James Harris === Subject: Re: Implicit differentiation help > Hi i also need help with an implicit differentiation question. The > question is to Žn the 2nd derivative as a function of x if sin y+cos > y=x. I can get to a Žnal answer but it i cant get rid of the y > values i have there. Could anyonehelp me get an answer in terms of I think this problem can be attacked primarily by going through the motions: (1) sin(y) + cos(y) = x Differentiate w.r.t. x: (2) [cos(y) - sin(y)] * y¹ = 1 Multiply (2) by 1/y¹ to get: (3) cos(y) - sin(y) = 1/y¹ Now square (1) and (3) to get: (4) sin^2(y) + 2*sin(y)*cos(y) + cos^2(y) = x^2 (5) sin^2(y) - 2*sin(y)*cos(y) + cos^2(y) = (1/y¹)^2 The previous reorganization is a standard trick, so take note. Now add (4) and (5) to get: (6) 2 = x^2 + (1/y¹)^2 Isolate y¹: (7) y¹ = sqrt[1/(2 - x^2)] It¹s your job to Žnish. Note that the square root can be positive or negative. There are other ways of arriving at this equation, but this one is cleaner that the Žrst one I thought up. You should be able to proceed from here. Hope I didn¹t do too much of your homework for you! .... Bob === Subject: Probability Question from a Computer Course BACKGROUND: part that modeled this situation (a sample output from that part is at the end of this post): There are n=10 mines in the water. A sonar can send one ping at a time, (which will reach all n mines). The ping will detect an individual mine with a probability p=0.05. Once a mine is detected, it stays detected forever (don¹t need to Žnd it again, although a later ping may or may not detect it). How long will it take to discover all 10 mines? (The assignment had some other big parts to it; I didn¹t write it to Žnd the repeated average of this part. I ran a few trials and came up with 67, 49, 46, 65, and 67 pings). QUESTION: How would one compute the actual expected value of the number of pings required? This was not a part of the assignment, but has been bugging me since I¹m also in a beginning probability course. My thinking so far: 1. The expected number of detections in each ping is 1/2 a mine, but that doesn¹t seem to lead anywhere. 2. Each ping is a binomial process with n=10, p=0.05. So on each ping, I could calculate the probability of exactly r=0 detections, r=1 detections, etc. 3. Now I have the probabilities of the 11 possible outcomes (detected r=0 mines, r=1, ... ,r=10) of the Žrst ping from 2. I could calculate the conditional probabilities of the outcomes of the second ping: 3.a. Given r=0 mines detected on the Žrst ping, repeat 2. 3.b. Given r=1 mines detected on the Žrst ping, calculate the 10 binomial probabilities of r=0..9 detections, with n=9, p=0.05. 3.c. Given r=2..r=10 on the Žrst ping, similar to 3.b. So, I would do something like 10! (or is it 10^10?) computations of the binomial formula, summing expected values along each branch of a huge tree. There must be a nicer way. Any suggestions? *Sample Output of my program* --------------------------------------- Creating mine #0 Creating mine #1 Creating mine #2 Creating mine #3 Creating mine #4 Creating mine #5 Creating mine #6 Creating mine #7 Creating mine #8 Creating mine #9 Ping #1 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: false mySonar pinged mine #3: no contact Has been detected: false mySonar pinged mine #4: no contact Has been detected: false mySonar pinged mine #5: no contact Has been detected: false mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: CONTACT! Has been detected: true mySonar pinged mine #8: no contact Has been detected: false mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 1 Ping #2 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: false mySonar pinged mine #3: no contact Has been detected: false mySonar pinged mine #4: no contact Has been detected: false mySonar pinged mine #5: no contact Has been detected: false mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: false mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 1 Ping #3 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: false mySonar pinged mine #3: no contact Has been detected: false mySonar pinged mine #4: no contact Has been detected: false mySonar pinged mine #5: no contact Has been detected: false mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: CONTACT! Has been detected: true mySonar pinged mine #8: no contact Has been detected: false mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 1 Ping #4 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: false mySonar pinged mine #3: no contact Has been detected: false mySonar pinged mine #4: no contact Has been detected: false mySonar pinged mine #5: no contact Has been detected: false mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: false mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 1 Ping #5 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: false mySonar pinged mine #3: no contact Has been detected: false mySonar pinged mine #4: no contact Has been detected: false mySonar pinged mine #5: no contact Has been detected: false mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: false mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 1 Ping #6 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: false mySonar pinged mine #3: no contact Has been detected: false mySonar pinged mine #4: no contact Has been detected: false mySonar pinged mine #5: no contact Has been detected: false mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: false mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 1 Ping #7 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: false mySonar pinged mine #3: no contact Has been detected: false mySonar pinged mine #4: no contact Has been detected: false mySonar pinged mine #5: no contact Has been detected: false mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: false mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 1 Ping #8 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: false mySonar pinged mine #3: no contact Has been detected: false mySonar pinged mine #4: no contact Has been detected: false mySonar pinged mine #5: no contact Has been detected: false mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: false mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 1 Ping #9 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: false mySonar pinged mine #3: CONTACT! Has been detected: true mySonar pinged mine #4: no contact Has been detected: false mySonar pinged mine #5: no contact Has been detected: false mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: false mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 2 Ping #10 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: false mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: false mySonar pinged mine #5: no contact Has been detected: false mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: false mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 2 Ping #11 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: false mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: false mySonar pinged mine #5: no contact Has been detected: false mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: false mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 2 Ping #12 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: false mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: false mySonar pinged mine #5: no contact Has been detected: false mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: false mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 2 Ping #13 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: false mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: false mySonar pinged mine #5: no contact Has been detected: false mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: false mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 2 Ping #14 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: false mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: false mySonar pinged mine #5: CONTACT! Has been detected: true mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: false mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 3 Ping #15 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: false mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: false mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: false mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 3 Ping #16 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: false mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: false mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: CONTACT! Has been detected: true mySonar pinged mine #8: no contact Has been detected: false mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 3 Ping #17 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: CONTACT! Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: false mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: CONTACT! Has been detected: true mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 5 Ping #18 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: false mySonar pinged mine #5: CONTACT! Has been detected: true mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 5 Ping #19 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: false mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 5 Ping #20 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: CONTACT! Has been detected: true mySonar pinged mine #4: no contact Has been detected: false mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 5 Ping #21 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: CONTACT! Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: CONTACT! Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 6 Ping #22 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: CONTACT! Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: CONTACT! Has been detected: true mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 6 Ping #23 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 6 Ping #24 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 6 Ping #25 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 6 Ping #26 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 6 Ping #27 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 6 Ping #28 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 6 Ping #29 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 6 Ping #30 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 6 Ping #31 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: CONTACT! Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 6 Ping #32 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 6 Ping #33 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 6 Ping #34 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 6 Ping #35 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 6 Ping #36 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: CONTACT! Has been detected: true mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 6 Ping #37 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 6 Ping #38 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 6 Ping #39 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: false Total number of objects detected by SONAR so far = 6 Ping #40 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: CONTACT! Has been detected: true Total number of objects detected by SONAR so far = 7 Ping #41 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: true Total number of objects detected by SONAR so far = 7 Ping #42 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: true Total number of objects detected by SONAR so far = 7 Ping #43 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: true Total number of objects detected by SONAR so far = 7 Ping #44 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: false mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: true Total number of objects detected by SONAR so far = 7 Ping #45 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: CONTACT! Has been detected: true mySonar pinged mine #5: CONTACT! Has been detected: true mySonar pinged mine #6: CONTACT! Has been detected: true mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: true Total number of objects detected by SONAR so far = 8 Ping #46 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: CONTACT! Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: true mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: true Total number of objects detected by SONAR so far = 8 Ping #47 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: CONTACT! Has been detected: true mySonar pinged mine #6: no contact Has been detected: true mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: true Total number of objects detected by SONAR so far = 8 Ping #48 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: CONTACT! Has been detected: true mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: true Total number of objects detected by SONAR so far = 8 Ping #49 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: true mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: true Total number of objects detected by SONAR so far = 8 Ping #50 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: true mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: true Total number of objects detected by SONAR so far = 8 Ping #51 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: true mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: true Total number of objects detected by SONAR so far = 8 Ping #52 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: true mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: true Total number of objects detected by SONAR so far = 8 Ping #53 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: false mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: true mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: true Total number of objects detected by SONAR so far = 8 Ping #54 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: CONTACT! Has been detected: true mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: CONTACT! Has been detected: true mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: true Total number of objects detected by SONAR so far = 9 Ping #55 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: true mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: true mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: CONTACT! Has been detected: true mySonar pinged mine #9: no contact Has been detected: true Total number of objects detected by SONAR so far = 9 Ping #56 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: CONTACT! Has been detected: true mySonar pinged mine #2: CONTACT! Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: true mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: true Total number of objects detected by SONAR so far = 9 Ping #57 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: true mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: true mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: true Total number of objects detected by SONAR so far = 9 Ping #58 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: true mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: true mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: CONTACT! Has been detected: true mySonar pinged mine #9: no contact Has been detected: true Total number of objects detected by SONAR so far = 9 Ping #59 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: true mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: CONTACT! Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: true mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: true Total number of objects detected by SONAR so far = 9 Ping #60 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: true mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: CONTACT! Has been detected: true mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: true Total number of objects detected by SONAR so far = 9 Ping #61 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: true mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: CONTACT! Has been detected: true mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: true Total number of objects detected by SONAR so far = 9 Ping #62 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: true mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: true mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: true Total number of objects detected by SONAR so far = 9 Ping #63 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: true mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: true mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: true Total number of objects detected by SONAR so far = 9 Ping #64 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: true mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: true mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: true Total number of objects detected by SONAR so far = 9 Ping #65 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: true mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: true mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: CONTACT! Has been detected: true mySonar pinged mine #9: no contact Has been detected: true Total number of objects detected by SONAR so far = 9 Ping #66 mySonar pinged mine #0: no contact Has been detected: false mySonar pinged mine #1: no contact Has been detected: true mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: true mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: true Total number of objects detected by SONAR so far = 9 Ping #67 mySonar pinged mine #0: CONTACT! Has been detected: true mySonar pinged mine #1: no contact Has been detected: true mySonar pinged mine #2: no contact Has been detected: true mySonar pinged mine #3: no contact Has been detected: true mySonar pinged mine #4: no contact Has been detected: true mySonar pinged mine #5: no contact Has been detected: true mySonar pinged mine #6: no contact Has been detected: true mySonar pinged mine #7: no contact Has been detected: true mySonar pinged mine #8: no contact Has been detected: true mySonar pinged mine #9: no contact Has been detected: true Total number of objects detected by SONAR so far = 10 === Subject: Re: Probability Question from a Computer Course >BACKGROUND: >part that modeled this situation (a sample output from that part is at >the end of this post): >There are n=10 mines in the water. >A sonar can send one ping at a time, (which will reach all n mines). >The ping will detect an individual mine with a probability p=0.05. >Once a mine is detected, it stays detected forever (don¹t need to >Žnd it again, although a later ping may or may not detect it). >How long will it take to discover all 10 mines? >(The assignment had some other big parts to it; I didn¹t write it to >Žnd the repeated average of this part. I ran a few trials and came up >with 67, 49, 46, 65, and 67 pings). >QUESTION: >How would one compute the actual expected value of the number of pings >required? This was not a part of the assignment, but has been bugging >me since I¹m also in a beginning probability course. You¹ll want to check all of the calculations, as I haven¹t had time to do much double-checking. For later convenience let q = 1 - p. Let p(k,r) be the probability that exactly r mines remain undetected after the Žrst k pings. There are C(n,r) ways of choosing r mines. The probability that a particular mine remains undetected after k pings is q^k, so the probability that every mine in a particular set of r mines remains undetected after k pings is (q^k)^r = q^(rk). The probability that a given mine is detected in at most k pings is 1 - q^k, so the probability that every mine in a particular set of n - r mines is detected within k pings is (1 - q^k)^(n-r). Thus, p(k,r) = C(n,r)*q^(kr)*(1 - q^k)^(n-r). In particular, p(k,0) = (1 - q^k)^n (which reassuringly does approach 1 as k increases!). This isn¹t the probability that it will take k pings to detect all n mines, however, but rather the probability that it will take at most k pings. The probability, P(k), that it will take exactly k pings must be P(k) = p(k,0) - p(k-1,0) = (1 - q^k)^n - (1 - q^(k-1))^n, and e, the expected number of steps, must be e = sum{k > 0; k[(1 - q^k)^n - (1 - q^(k-1))^n]}. Consider the m-th partial sum e(m): e(m) = sum{k=1 to m; k[(1 - q^k)^n - (1 - q^(k-1))^n]} = sum{k=1 to m; k(1 - q^k)^n} - sum{k=1 to m; k(1 - q^(k-1))^n} = sum{k=1 to m; k(1 - q^k)^n} - sum{k=2 to m; k(1 - q^(k-1))^n} = sum{k=1 to m; k(1 - q^k)^n} - sum{k=1 to m-1; (k + 1)(1 - q^k)^n} = sum{k=1 to m; k(1 - q^k)^n} - sum{k=1 to m-1; k(1 - q^k)^n} - sum{k=1 to m-1; (1 - q^k)^n} = m(1 - q^m)^n - sum{k=1 to m-1; (1 - q^k)^n}. Now sum{k=1 to m-1; (1 - q^k)^n} = sum{k=1 to m-1; sum{j; C(n,j) * (-1)^j * q^(jk)}} = sum{j=0 to n; (-1)^j * C(n,j)*sum{k=1 to m-1; q^(jk)}} = (m - 1) + sum{j=1 to n; (-1)^j * C(n,j)*sum{k=1 to m-1; q^(jk)}} = (m - 1) + sum{j=0 to n; (-1)^j * C(n,j) * (q^j - q^jm)/(1 - q^j)}. Similarly, m(1 - q^m)^n = m + sum{j=1 to n; m * C(n,j) * (-1)^j * q^(jm)}, so e(m) = 1 + sum{j=1 to n; C(n,j)*(-1)^j* [q^(jm)} - (q^j - q^(jm))/(1 - q^j]}. Taking the limit as m increases without bound, we get e = 1 - sum{j=1 to n; C(n,j) * (-1)^j * q^j / (1 - q^j)}. [*] As a rough check, note that when n = 1 this reduces to e = 1 + q/(1 - q) = 1/(1 - q) = 1/p, as expected. I don¹t immediately see a nice closed form for e in [*], but this at least reduces the problem considerably. Brian === Subject: Re: addition > I need help with adding two numbers (like 1, 2, and maybe even > others). How¹s it possible to do it without a calculator?! Also, what > does it mean when the book says 1 + 1 + 1? I¹ve only heard about > adding two numbers, not three. My calculus teacher wasn¹t very clear > about it. It is considered an abomination to add a number other than 1 to another number. So if you are confronted with say 50 + 20 you must break one of these larger numbers down to a series of 1s to get back to basic deŽnitions. (Hint, choose the smaller one.) So we get 50 + (1 + 1 + 1 ..... +1) You move the Žrst 1 outside the parentheses and you get 51 + (1+1+ .... +1) You keep going like this one at a time and Žnally you will end up with 69 + 1 = 70. The general result is known as the ones theorem. You must always apply the ones theroem whenever you have numbers to add together. The reason your calculus teacher would not talk about adding three numbers is that is also considered an abomination. This all goes back to Noah¹s Ark where the animals were in twos - never threes. However, you can achieve the same result by applying the ones theorem in succession. You should try to teach others about this, but you may have better luck outside of school where people have not been contaminated by false ideas. For example, if you are at a truck stop on the way to University and a waiter is adding more than two numbers and not applying the ones therom you should tell him about the ones theorem and that what he is doing is an abomination. Be very persistant and I guarrantee you at the end there will be a conversion. Bill === Subject: Re: addition >> I need help with adding two numbers (like 1, 2, and maybe even >> others). How¹s it possible to do it without a calculator?! Also, what >> does it mean when the book says 1 + 1 + 1? I¹ve only heard about >> adding two numbers, not three. My calculus teacher wasn¹t very clear >> about it. > It is considered an abomination to add a number other than 1 to another > number. So if you are confronted with say 50 + 20 you must break one of these > larger numbers down to a series of 1s to get back to basic deŽnitions. (Hint, > choose the smaller one.) So we get 50 + (1 + 1 + 1 ..... +1) You move the > Žrst 1 outside the parentheses and you get 51 + (1+1+ .... +1) You keep going > like this one at a time and Žnally you will end up with 69 + 1 = 70. The > general result is known as the ones theorem. You must always apply the ones > theroem whenever you have numbers to add together. Is this a consequence of the axioms that natural numbers, have just read them, or are you just messing with the OP? > The reason your calculus teacher would not talk about adding three numbers is > that is also considered an abomination. This all goes back to Noah¹s Ark where > the animals were in twos - never threes. However, you can achieve the same > result by applying the ones theorem in succession. > You should try to teach others about this, but you may have better luck > outside of school where people have not been contaminated by false ideas. For > example, if you are at a truck stop on the way to University and a waiter is > adding more than two numbers and not applying the ones therom you should tell > him about the ones theorem and that what he is doing is an abomination. Be > very persistant and I guarrantee you at the end there will be a conversion. > Bill -- Sigblock empty. By choice. === Subject: Re: addition > Is this a consequence of the axioms that natural numbers, have just read > them, or are you just messing with the OP? Not all or¹s are exclusive. Bill === Subject: Re: addition >I don¹t know who you¹re trying to impress, but there¹s no way you can be in >calculus class and not know how to evaluate 1+1+1. You¹re probably right. On the other hand, lots of my calculus (and statistics) students have been unable to evaluate sums like 37/1000 + 43/1000 correctly. The answer 80/2000 is distressingly common. -- Stan Brown, Oak Road Systems, Cortland County, New York, USA http://OakRoadSystems.com Address munging may or may not reduce the spam you get; it surely reduces the number of useful answers you get. http://www.cs.tut.Ž/~jkorpela/usenet/laws.html === Subject: diagonalization by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9J2xcs05161; The previous poster gave some historical remarks but did not answer the original poster¹s question. The diagonalization approach will fail for rationals because the new number created by altering the i¹th digit of each i¹th listed decimal will be irrational, not rational. For proofs of the countability of the rationals, see, for instance, Schaum¹s Outline of Set Theory. --g === Subject: getting rid of y by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9J2xcK05157; One option would be to use a trig identity: Since you have x = sin(y) + cos(y), squaring both sides yields: x^2 = sin^2(y)+cos^2(y)+2sin(y)cos(y). Now, since sin^2(y)+cos^2(y)=1, you have: x^2-1 = 2sin(y)cos(y). By a trig identity, the right hand side can be re-written as sin(2y); so you have: x^2-1=sin(2y), and now you can obtain y as an explicit function of x by taking inverse sin of both sides. === Subject: Re: getting rid of y > One option would be to use a trig identity: Since you have > x = sin(y) + cos(y), squaring both sides yields: > x^2 = sin^2(y)+cos^2(y)+2sin(y)cos(y). Now, since > sin^2(y)+cos^2(y)=1, you have: > x^2-1 = 2sin(y)cos(y). By a trig identity, the right hand side can be > re-written as sin(2y); so you have: > x^2-1=sin(2y), and now you can obtain y as an explicit function of x > by taking inverse sin of both sides. Slick. === Subject: Help on a proof for convergence? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9J2xb805149; How would one go about proving the convergence of the following series: sin(n)/n ? There is no interval given. Most of the tests used for proving convergence cannot be used: alternating series (because the terms are not all decreasing, as is the case if it was only 1 to inŽnity), the comparison tests (you can¹t compare the series to 1/n), and the root and ratio tests simply make the series more difŽcult. So does anyone know how to prove this? Any help would be appreciated. === Subject: Re: Help on a proof for convergence? > How would one go about proving the convergence of the following > series: > sin(n)/n > ? There is no interval given. Most of the tests used for proving > convergence cannot be used: alternating series (because the terms are > not all decreasing, as is the case if it was only 1 to inŽnity), the > comparison tests (you can¹t compare the series to 1/n), and the root > and ratio tests simply make the series more difŽcult. So does anyone > know how to prove this? Any help would be appreciated. Hi Johnny, nice question ! I understand that you need the convergence of S:= SUM_{k=1 to k= infty}sin(k)/k . Try Fourier series of certain odd function f(x) such that f(x)= SUM_{k=1 to k=infty} sin(kx)/k . Then , if possible put x=1. Perhaps help. Alex Note that S_n(x):=SUM_{k=1 to k=n} sin(kx)/k >0 for all x in (0,pi) . This is Fejer¹s inequality (1905 ?) . I think that you must take the 2*pi periodic odd function f:R-->R whose restriction at (-pi,pi] is deŽned as f(x)= (x-pi)/pi for x in (-pi ,0) f(x)= 0 when x=0 f(x)= (pi-x)/pi for x in (0,pi] . If you look in a treatise on trigonometric series (A.Zygmund or ?.Edwards) you will see that the convergence is assured. Therefore, my impresion is that the sum of your series is S=f(1)=(pi-1)/pi = 1- 1/pi . === Subject: Re: Help on a proof for convergence? Not helpful on any proofs, but you may Žnd this link interesting... http://mathworld.wolfram.com/SincFunction.html -- Dana = = = = = = = = = = = = = = = = = > How would one go about proving the convergence of the following > series: > sin(n)/n > ? There is no interval given. Most of the tests used for proving > convergence cannot be used: alternating series (because the terms are > not all decreasing, as is the case if it was only 1 to inŽnity), the > comparison tests (you can¹t compare the series to 1/n), and the root > and ratio tests simply make the series more difŽcult. So does anyone > know how to prove this? Any help would be appreciated. === Subject: Re: Help on a proof for convergence? >How would one go about proving the convergence of the following >series: >sin(n)/n >? There is no interval given. Most of the tests used for proving >convergence cannot be used: alternating series (because the terms are >not all decreasing, as is the case if it was only 1 to inŽnity), the >comparison tests (you can¹t compare the series to 1/n), Can¹t you??? Obviously the denominators are equal. Can¹t you compare the numerators? -1 < sin(n) < 1 for any integral n. N.B. <, not <=. You know this because the only time sin(x) = +1 or -1 is when x = k(pi/2)+1 for integer k. Since pi itself is irrational, k(pi/2)+1 can never be an integer; therefore sin(n) for integer n is never equal to +1 or -1. -- Stan Brown, Oak Road Systems, Cortland County, New York, USA http://OakRoadSystems.com Address munging may or may not reduce the spam you get; it surely reduces the number of useful answers you get. http://www.cs.tut.Ž/~jkorpela/usenet/laws.html === Subject: Re: Help on a proof for convergence? >>How would one go about proving the convergence of the following >>series: Sorry, I misread that as _sequence_. Please disregard my previous follow-up. -- Stan Brown, Oak Road Systems, Cortland County, New York, USA http://OakRoadSystems.com Address munging may or may not reduce the spam you get; it surely reduces the number of useful answers you get. http://www.cs.tut.Ž/~jkorpela/usenet/laws.html === Subject: Re: Help on a proof for convergence? > How would one go about proving the convergence of the following > series: > sin(n)/n > ? There is no interval given. I don¹t understand. Do you mean that the initial and Žnal values of the index n in the summation where not speciŽed? If so, then perhaps you were to assume that n goes from 1 to inŽnity. (If that¹s not true, then more information is needed.) > Most of the tests used for proving convergence cannot be used: > alternating series (because the terms are not all decreasing, as is the > case if it was only 1 to inŽnity), Right (despite the fact that MadJock seems to think that the terms decrease). > the comparison tests (you can¹t compare the series to 1/n), and the root > and ratio tests simply make the series more difŽcult. So does anyone > know how to prove this? Any help would be appreciated. If n goes from 1 to inŽnity, the series converges to (pi - 1)/2. That series arises fairly frequently in math newsgroups. Do a Google Groups search for sin(n)/n series convergence Of course you¹ll get some stuff which isn¹t pertinent, but you¹ll also get what you need. The second item I found was an excellent response by Rob Johnson, for example. BTW, if perchance the summation was instead supposed to be over _all_ integers n (from -inŽnity to +inŽnity), then you would most likely be supposed to take sin(n)/n to be 1 when n = 0, and the sum would then be, neatly, just pi. David Cantrell === Subject: Re: Help on a proof for convergence? === Subject: Help on a proof for convergence? >How would one go about proving the convergence of the following >series: sin(n)/n lim(n->oo) (sin n)/n = 0 Proof: For eps > 0, Žnd n0 with 1/n0 < eps Thus for all n > n0, |(sin n)/n - 0 | = |(sin n)/n| <= 1/n < 1/n0 < eps ---- === Subject: Re: Help on a proof for convergence? === > Subject: Help on a proof for convergence? > >How would one go about proving the convergence of the following > >series: sin(n)/n It¹s obvious that the _sequence_ sin(n)/n converges. But notice the word series above, William. Establishing convergence of the series is not so simple. David > lim(n->oo) (sin n)/n = 0 > Proof: > For eps > 0, Žnd n0 with 1/n0 < eps > Thus for all n > n0, > |(sin n)/n - 0 | = |(sin n)/n| <= 1/n < 1/n0 < eps === Subject: Re: Help on a proof for convergence? > >How would one go about proving the convergence of the following > >series: sin(n)/n > It¹s obvious that the _sequence_ sin(n)/n converges. But notice the word > series above, William. Establishing convergence of the series is not so > simple. Oh, OP wants to see the series sum_n (sin n)/n converge. Well now that that¹s unambigious. N modulus 2pi, is dense in the reals [0,2pi). Thus on the average it¹s an alternating series which makes it convergent, and also complicated. === Subject: Re: Help on a proof for convergence? Weird. Seems quite obvious in my head that it converges too - it¹s a common function in digital transmission systems. As n increases, f(n) decreases. But I don¹t know how to prove it. Sorry. MadJock > How would one go about proving the convergence of the following > series: > sin(n)/n > ? There is no interval given. Most of the tests used for proving > convergence cannot be used: alternating series (because the terms are > not all decreasing, as is the case if it was only 1 to inŽnity), the > comparison tests (you can¹t compare the series to 1/n), and the root > and ratio tests simply make the series more difŽcult. So does anyone > know how to prove this? Any help would be appreciated. === Subject: how would 4 9¹s equal 100 using basic math calculations? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9J62wC16584; I would like to ask for your help in solving a mathematical problem that has me stumped. Lets say you have four 9¹s. Now, using any basic arithmetical procedure (multiply, add, divide, subtract), how would you get them to equal 100. Is this possible or would this be a trick question? Any help would be greatly appreciated. Eric === Subject: Update to problem, my actual work done by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9JBpEV07258; I would like to apologize to everybody for not posting the work I actually did to attempt to solve this problem. Here is what I have done so far. Unless this is a trick question, or my math is poor, does this have a solution? Addition 9 + 9 = 18 + 9 = 27 + 9 = 36 9 + 9 = 18 + 9 = 27 - 9 = 18 9 + 9 = 18 + 9 = 27 * 9 = 243 9 + 9 = 18 + 9 = 27 / 9 = 3 9 + 9 = 18 - 9 = 9 + 9 = 18 9 + 9 = 18 - 9 = 9 - 9 = 0 9 + 9 = 18 - 9 = 9 * 9 = 81 9 + 9 = 18 - 9 = 9 / 9 = 1 9 + 9 = 18 * 9 = 169 + 9 = 176 9 + 9 = 18 * 9 = 169 - 9 = 160 9 + 9 = 18 * 9 = 169 * 9 = 1521 9 + 9 = 18 * 9 = 169 / 9 = 18.77777 9 + 9 = 18 / 9 = 9 + 9 = 18 9 + 9 = 18 / 9 = 9 - 9 = 0 9 + 9 = 18 / 9 = 9 * 9 = 81 9 + 9 = 18 / 9 = 9 / 9 = 1 Subtraction 9 - 9 = 0 + 9 = 9 + 9 = 18 9 - 9 = 0 + 9 = 9 - 9 = 0 9 - 9 = 0 + 9 = 9 * 9 = 81 9 - 9 = 0 + 9 = 9 / 9 = 1 9 - 9 = 0 - 9 = -9 + 9 = 0 9 - 9 = 0 - 9 = -9 - 9 = -18 9 - 9 = 0 - 9 = -9 * 9 = -81 9 - 9 = 0 - 9 = -9 / 9 = -1 9 - 9 = 0 * 9 = 0 + 9 = 9 9 - 9 = 0 * 9 = 0 - 9 = -9 9 - 9 = 0 * 9 = 0 * 9 = 0 9 - 9 = 0 * 9 = 0 / 9 = 0 9 - 9 = 0 / 9 = 0 + 9 = 9 9 - 9 = 0 / 9 = 0 - 9 = -9 9 - 9 = 0 / 9 = 0 * 9 = 81 9 - 9 = 0 / 9 = 0 / 9 = 9 Multiplication 9 * 9 = 81 + 9 = 90 + 9 = 99 9 * 9 = 81 + 9 = 90 - 9 = 81 9 * 9 = 81 + 9 = 90 * 9 = 810 9 * 9 = 81 + 9 = 90 / 9 = 10 9 * 9 = 81 - 9 = 72 + 9 = 81 9 * 9 = 81 - 9 = 72 - 9 = 63 9 * 9 = 81 - 9 = 72 * 9 = 648 9 * 9 = 81 - 9 = 72 / 9 = 8 9 * 9 = 81 * 9 = 729 + 9 = 738 9 * 9 = 81 * 9 = 729 - 9 = 720 9 * 9 = 81 * 9 = 729 * 9 = 6561 9 * 9 = 81 * 9 = 729 / 9 = 81 9 * 9 = 81 / 9 = 9 + 9 = 18 9 * 9 = 81 / 9 = 9 - 9 = 0 9 * 9 = 81 / 9 = 9 * 9 = 81 9 * 9 = 81 / 9 = 9 / 9 = 1 Division 9 / 9 = 1 + 9 = 10 + 9 = 19 9 / 9 = 1 + 9 = 10 - 9 = 1 9 / 9 = 1 + 9 = 10 * 9 = 90 9 / 9 = 1 + 9 = 10 / 9 = 1.1111 9 / 9 = 1 - 9 = -8 + 9 = 1 9 / 9 = 1 - 9 = -8 - 9 = -17 9 / 9 = 1 - 9 = -8 * 9 = -72 9 / 9 = 1 - 9 = -8 / 9 = -0.8888 9 / 9 = 1 * 9 = 9 + 9 = 18 9 / 9 = 1 * 9 = 9 - 9 = 0 9 / 9 = 1 * 9 = 9 * 9 = 81 9 / 9 = 1 * 9 = 9 / 9 = 1 9 / 9 = 1 / 9 = 0.1111 + 9 = 9.1111 9 / 9 = 1 / 9 = 0.1111 - 9 = -8.8888 9 / 9 = 1 / 9 = 0.1111 * 9 = 1 9 / 9 = 1 / 9 = 0.1111 / 9 = 0.0123456790 Eric === Subject: Re: how would 4 9¹s equal 100 using basic math calculations? > I would like to ask for your help in solving a mathematical problem > that has me stumped. Lets say you have four 9¹s. Now, using any basic > arithmetical procedure (multiply, add, divide, subtract), how would > you get them to equal 100. Is this possible or would this be a trick > question? 99 + 9/9 === by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9JBpDM07210; - when you square both sides of the original equation how do you end up with 2sin(y)cos(y) as I cannot see where that has come from - probably due to ignorance or the fact i have overlooked something. One other note just to be sure yogi how did the square¹s disappear from sin(y)y¹^2 and cos(y)y¹^2 when u took them across to the other side - i just need these points to check that i can go through the process === >- when you square both sides of the original equation how do you end >up with 2sin(y)cos(y) as I cannot see where that has come from - Neither can I, because of the way you posted. Much as I¹d like to help, I can¹t. When you¹re posting a follow-up, post it _as_ a follow-up. Your you¹re referring to. -- Stan Brown, Oak Road Systems, Cortland County, New York, USA http://OakRoadSystems.com Address munging may or may not reduce the spam you get; it surely reduces the number of useful answers you get. http://www.cs.tut.Ž/~jkorpela/usenet/laws.html === Subject: Dont worry about the Žnding 2sinycosy bit by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9JBpDZ07218; I was just being stupid - forgot for a moment that the whole thing was squared rather than the individual bits === Subject: Re: Quick Math Guide to core error issues James, You post so much material that I don¹t have the time or desire to read everything and I lose track of your position and arguments. Some time ago you had an organised website with deŽnitions and cross- reference and you have posted several summaries and histories on this newsgroup which would be easier to refer to if they were permanently available. I, for one, would be pleased if you would resurrect your website and put the information back. Here are just some suggestions as to what you could put there and how it could be organised. *DeŽnitions* - Algebraic Integers and Algebraic Numbers An explanation of what an algebraic integer is , some examples and their general properties would be useful. I vaguely recall that you had some software that would take a monic quadratic equation with integer coefŽcients and factorise it into linear terms with algebraic integer coefŽcients. This could usefully go here as a Java applet. - Your ŒObject Ring¹ and what numbers are in it and what aren¹t. I *think* you said 2^sqrt(3) was in it I *think* you said that it wasn¹t wholly contained in the complex numbers An explanation of why these numbers are in it would be useful - What an incomplete ring is. - Some expressions don¹t seem to translate across the Atlantic and I am thinking of Œhas a factor of 5¹ in particular. To me, Œhas a value of 42¹ means Œ42 is a value¹ so Œhas a factor of 5¹ should mean Œ5 is a factor¹ but you don¹t seem to mean this all time. Personally, I prefer the active Œ5 divides N¹ or perhaps ŒN is divisible by 5¹ to the passive ŒN has 5 as a factor¹. *History* I think you have written at least two histories to this newsgroup and a web site would be a much better place to put them. You have made several accusations of lying and you could use the website to substantiate them by pointing to documentation. *Points of Contention* Your Viewpoint I lose track of the arguments and how they are settled or even if they are. Your executive summary below is a good start but that is all it is. Hotlinks to expansions, with proofs, would be better. Java documentation is a fairly good example if the analogy isn¹t taken too far - at the top of the documentation for a class is an explanation of what this class is for and a list of the methods with the types of their parameters and each has a hotlink to an explanation of that method. One could go further and look at the code but it should never be necessary. Similarly, a mathematical exposition can have deŽnitions, lemmas, propositions and theorems. The proof of a lemma or similar can cite consequences from the statement of another but mustn¹t ever refer to the proof - this would be as bad as code jumping into the middle of a subroutine or the documentation of a method referring to the code of another method. I suggest an exposition of your work would have the same structure with an executive summary at the top and cascading expansions of each point. (A good book contains a series of chapters. Each chapter starts out by saying what it is going to say, then says it in detail, and Žnally ends by summarising what it has said) Your previous website made a point of being terse but this isn¹t necessary. Other Viewpoints It is not at all an easy thing to do, but if you could try to explain, in detail, other people¹s standpoints as well as your own then this would be useful. I¹m thinking in particular of the factorisation of certain polynomials in which you say that one root is coprime to a prime factor of the constant term of the polynomial and other people have shown explicit factorisations of the polynomial in which this is not true. I didn¹t follow your rebuttal. Dead Viewpoints It would take great courage but if you could also document points where you are no longer holding a former position then this also would interesting. I¹m thinking in particular of whether a ring must be closed under an inŽnite summation and whether Z[pi] is isomorphic to the reals. *SimpliŽcations* The polynomial that you use is very complicated with several parameters. Is all the complication necessary to the proof and does it add to it? Is the Œu¹ term necessary at all? You have resorted to actually substituting values in order to explain - why not go back and give the whole argument again but use a polynomial which is as simple as possible. In fact, I don¹t always follow your reasoning but perhaps it would be more obvious if you used a quadratic polynomial instead of a trinomial or explained why the argument doesn¹t work for quadratics. *Style* I, too, like long sentences (my second choice for a book to take if I were to be stranded on a desert island would be the collected works of Jane Austen) and you are much better at writing them than I am but they can be difŽcult for other people to follow. Take this as an example: . I already have as the polynomial is . . P(m) = f^2((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1+mf^2 )x u^2 + u^3f) . . and your attempts at confusing the issue by trying to push up x, . don¹t help you here, as the g¹s as factors of m, necessarily have . a constant term with respect to m, and your claim that it can vary . with m, is nonsensical on its face. It makes sense and I can understand it but I have to work at it. I suggest that shorter sentences would make it easier for the reader. Penny Hassett > For those of you trying to keep up with the mathematical facts in the > discussions about the error in core mathematics from a problem with a > deŽnition, this post will outline the important ones quickly and > succinctly. > 1. First the problematic deŽnition: > Algebraic integers are deŽned to be roots of monic polynomials with > integer coefŽcient e.g. x^3 + 3x + 1 or x^234 - 34x^12 + 17, where > monic refers to the leading coefŽcient. > My assertion is that the over hundred year old deŽnition excludes > numbers that have to be included to keep from having contradiction > i.e. mathematical inconsistency. > 2. The important tool I use is a polynomial: > P(m) = f^2((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1+mf^2 )x u^2 + u^3 f) > The form of the polynomial allows me to factor P(m) into > non-polynomial factors, and the factorization with those factors is > P(m) = (a_1 x + uf)(a_2 x + uf)(a_3 x + uf) > where the a¹s are roots of the following cubic: > a^3 + 3(-1+mf^2)a^2 - f^2(m^3 f^4 - 3m^2 f^2 + 3m). > 3. Dispute centers around what happens when I divide P(m) by f^2, > which you¹ll note is a factor of the polynomial in the ring of > algebraic integers. > 4. Mathematicians have argued that f^2 divides off as a function of m > because if they concede that it divides off independent of m, then I > can show that only two of the roots of > a^3 + 3(-1+mf^2)a^2 - f^2(m^3 f^4 - 3m^2 f^2 + 3m) > have f as a factor. > 5. However, it turns out that if you go to the Želd of algebraic > numbers you can prove that for *certain* values of m and f, the roots > of the cubic do not have f as a factor *in the ring of algebraic > numbers* which is the inconsistency. > That is, for the math to be consistent, two of the roots *should* have > f as a factor as long as m and f are algebraic integers, but while I > can show they do for a particular values like m=1, f=sqrt(2), there > are other values you can show they do not *in the ring of algebraic > integers* which results from the deŽnition and its focus on monic > polynomials. > Note: In the ring of algebraic integers you can¹t see the problem but > have to go to the Želd of algebraic numbers as from within the ring > of algebraic integers it appears that only two of the roots have a > factor that is f. > James Harris === Subject: Re: Quick Math Guide to core error issues > James, > You post so much material that I don¹t have the time or desire to > read everything and I lose track of your position and arguments. > Some time ago you had an organised website with deŽnitions and cross- > reference and you have posted several summaries and histories on this > newsgroup which would be easier to refer to if they were permanently > available. I, for one, would be pleased if you would resurrect your > website and put the information back. > Here are just some suggestions as to what you could put there and > how it could be organised. > *DeŽnitions* > - Algebraic Integers and Algebraic Numbers > An explanation of what an algebraic integer is , some > examples and their general properties would be useful. > I vaguely recall that you had some software that would > take a monic quadratic equation with integer coefŽcients > and factorise it into linear terms with algebraic integer > coefŽcients. This could usefully go here as a Java applet. > - Your ŒObject Ring¹ and what numbers are in it and what aren¹t. > I *think* you said 2^sqrt(3) was in it > I *think* you said that it wasn¹t wholly contained in the > complex numbers > An explanation of why these numbers are in it would be useful > - What an incomplete ring is. > - Some expressions don¹t seem to translate across the Atlantic and > I am thinking of Œhas a factor of 5¹ in particular. To me, Œhas a > value of 42¹ means Œ42 is a value¹ so Œhas a factor of 5¹ should > mean Œ5 is a factor¹ but you don¹t seem to mean this all time. > Personally, I prefer the active Œ5 divides N¹ or perhaps ŒN is > divisible by 5¹ to the passive ŒN has 5 as a factor¹. > *History* > I think you have written at least two histories to this newsgroup > and a web site would be a much better place to put them. You have > made several accusations of lying and you could use the website > to substantiate them by pointing to documentation. > *Points of Contention* > Your Viewpoint > I lose track of the arguments and how they are settled or even > if they are. Your executive summary below is a good start but that > is all it is. Hotlinks to expansions, with proofs, would be better. > Java documentation is a fairly good example if the analogy isn¹t > taken too far - at the top of the documentation for a class is > an explanation of what this class is for and a list of the methods > with the types of their parameters and each has a hotlink to an > explanation of that method. One could go further and look at the > code but it should never be necessary. > Similarly, a mathematical exposition can have deŽnitions, lemmas, > propositions and theorems. The proof of a lemma or similar can cite > consequences from the statement of another but mustn¹t ever refer to > the proof - this would be as bad as code jumping into the middle of > a subroutine or the documentation of a method referring to the > code of another method. > I suggest an exposition of your work would have the same structure > with an executive summary at the top and cascading expansions of > each point. (A good book contains a series of chapters. Each chapter > starts out by saying what it is going to say, then says it in detail, > and Žnally ends by summarising what it has said) Your previous > website > made a point of being terse but this isn¹t necessary. > Other Viewpoints > It is not at all an easy thing to do, but if you could try to > explain, > in detail, other people¹s standpoints as well as your own then this > would be useful. I¹m thinking in particular of the factorisation > of certain polynomials in which you say that one root is coprime > to a prime factor of the constant term of the polynomial and other > people have shown explicit factorisations of the polynomial in > which this is not true. I didn¹t follow your rebuttal. > Dead Viewpoints > It would take great courage but if you could also document points > where you are no longer holding a former position then this also > would interesting. I¹m thinking in particular of whether a ring > must be closed under an inŽnite summation and whether Z[pi] is > isomorphic to the reals. > > *SimpliŽcations* > The polynomial that you use is very complicated with several > parameters. Is all the complication necessary to the proof and does > it add to it? Is the Œu¹ term necessary at all? You have resorted > to actually substituting values in order to explain - why not go > back and give the whole argument again but use a polynomial which > is as simple as possible. In fact, I don¹t always follow your > reasoning but perhaps it would be more obvious if you used a > quadratic polynomial instead of a trinomial or explained why the > argument doesn¹t work for quadratics. > *Style* > > I, too, like long sentences (my second choice for a book to take > if I were to be stranded on a desert island would be the collected > works of Jane Austen) and you are much better at writing them than > I am but they can be difŽcult for other people to follow. > Take this as an example: > . I already have as the polynomial is > . P(m) = f^2((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1+mf^2 )x u^2 + u^3f) > . and your attempts at confusing the issue by trying to push up x, > . don¹t help you here, as the g¹s as factors of m, necessarily have > . a constant term with respect to m, and your claim that it can vary > . with m, is nonsensical on its face. > It makes sense and I can understand it but I have to work at it. > I suggest that shorter sentences would make it easier for the > reader. > Penny Hassett > > For those of you trying to keep up with the mathematical facts in the > discussions about the error in core mathematics from a problem with a > deŽnition, this post will outline the important ones quickly and > succinctly. > > 1. First the problematic deŽnition: > > Algebraic integers are deŽned to be roots of monic polynomials with > integer coefŽcient e.g. x^3 + 3x + 1 or x^234 - 34x^12 + 17, where > monic refers to the leading coefŽcient. > > My assertion is that the over hundred year old deŽnition excludes > numbers that have to be included to keep from having contradiction > i.e. mathematical inconsistency. > > 2. The important tool I use is a polynomial: > > P(m) = f^2((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1+mf^2 )x u^2 + u^3 f) > > The form of the polynomial allows me to factor P(m) into > non-polynomial factors, and the factorization with those factors is > > P(m) = (a_1 x + uf)(a_2 x + uf)(a_3 x + uf) > > where the a¹s are roots of the following cubic: > > a^3 + 3(-1+mf^2)a^2 - f^2(m^3 f^4 - 3m^2 f^2 + 3m). > > 3. Dispute centers around what happens when I divide P(m) by f^2, > which you¹ll note is a factor of the polynomial in the ring of > algebraic integers. > > 4. Mathematicians have argued that f^2 divides off as a function of m > because if they concede that it divides off independent of m, then I > can show that only two of the roots of > > a^3 + 3(-1+mf^2)a^2 - f^2(m^3 f^4 - 3m^2 f^2 + 3m) > > have f as a factor. > > 5. However, it turns out that if you go to the Želd of algebraic > numbers you can prove that for *certain* values of m and f, the roots > of the cubic do not have f as a factor *in the ring of algebraic > numbers* which is the inconsistency. > > That is, for the math to be consistent, two of the roots *should* have > f as a factor as long as m and f are algebraic integers, but while I > can show they do for a particular values like m=1, f=sqrt(2), there > are other values you can show they do not *in the ring of algebraic > integers* which results from the deŽnition and its focus on monic > polynomials. > > Note: In the ring of algebraic integers you can¹t see the problem but > have to go to the Želd of algebraic numbers as from within the ring > of algebraic integers it appears that only two of the roots have a > factor that is f. > > James Harris What a breath of fresh air you are on this NG! Respectfully, John === Subject: Re: Quick Math Guide to core error issues > James, > You post so much material that I don¹t have the time or desire to > read everything and I lose track of your position and arguments. > Some time ago you had an organised website with deŽnitions and cross- > reference and you have posted several summaries and histories on this > newsgroup which would be easier to refer to if they were permanently > available. I, for one, would be pleased if you would resurrect your > website and put the information back. I like you Penny Hasset and appreciate your commentary which is why I¹m posting in a thread where I don¹t need to post as it¹s a quide. My problem is that I don¹t want to use MSN Groups, and I don¹t feel like going to another website provider. I *am* willing to allow someone else to host my work as long as they give me complete editorial control. Otherwise, it¹s easier for me to just post rather than try to maintain a website. > Here are just some suggestions as to what you could put there and > how it could be organised. > *DeŽnitions* > - Algebraic Integers and Algebraic Numbers > An explanation of what an algebraic integer is , some > examples and their general properties would be useful. > I vaguely recall that you had some software that would > take a monic quadratic equation with integer coefŽcients > and factorise it into linear terms with algebraic integer > coefŽcients. This could usefully go here as a Java applet. Oh yeah, after Arturo Magidin tried to make a big deal out of some crap, I Žgured out what he was doing, which wasn¹t much more than a rather simple search for a factorization in algebraic integers. It was fun, but not that much fun. You see, I¹ve found and dropped more mathematics than most people discover in a lifetime, as I¹m a thrill seeker. You know, an adrenaline junkie. > - Your ŒObject Ring¹ and what numbers are in it and what aren¹t. > I *think* you said 2^sqrt(3) was in it > I *think* you said that it wasn¹t wholly contained in the > complex numbers > An explanation of why these numbers are in it would be useful Actually *you* are in it Penny Hasset, as the object ring is rather large. You see, you are a mathematical object which I can prove using some rather basic logic and Goedel¹s proof. I like it that you¹re in Britain. If you¹re willing to advise me, I¹m willing to toe a line. After all, it is math, but by myself I tend to be over the top. I need organization. > - What an incomplete ring is. It¹s a ring where you can have contradictions *within* the ring, which is the problem with algebraic integers. I can explain everything, but I¹m a discoverer. I¹m an artist. I¹m NOT organized for this other stuff, like trying to convince people. I¹m an artist. > - Some expressions don¹t seem to translate across the Atlantic and > I am thinking of Œhas a factor of 5¹ in particular. To me, Œhas a > value of 42¹ means Œ42 is a value¹ so Œhas a factor of 5¹ should > mean Œ5 is a factor¹ but you don¹t seem to mean this all time. > Personally, I prefer the active Œ5 divides N¹ or perhaps ŒN is > divisible by 5¹ to the passive ŒN has 5 as a factor¹. Hey Penny Hasset, if you can help me, and help me overcome these objections to the extent that I can make some money here, I¹ll pay you $250,000 US from any one math prize that I win that exceeds that amount. Since I¹m a black male in America, as it has a rather stupendous history of racism, I should be able to pay that amount as well as $100,000 US as previously offered to a person or group with a machine proof of the core error, without much trouble, since I turn the world upside down. > *History* > I think you have written at least two histories to this newsgroup > and a web site would be a much better place to put them. You have > made several accusations of lying and you could use the website > to substantiate them by pointing to documentation. Oh, that¹s part of my fun. Unfortunately for me I have a tendency to scare people away when they Žgure out just how much I know and what I can do. Luckily for me, mathematicians are arrogant *and* dumb. So they¹re a perfect combination for someone like me, who otherwise gets kind of lonely. > *Points of Contention* > Your Viewpoint > I lose track of the arguments and how they are settled or even > if they are. Your executive summary below is a good start but that > is all it is. Hotlinks to expansions, with proofs, would be better. > Java documentation is a fairly good example if the analogy isn¹t > taken too far - at the top of the documentation for a class is > an explanation of what this class is for and a list of the methods > with the types of their parameters and each has a hotlink to an > explanation of that method. One could go further and look at the > code but it should never be necessary. > Similarly, a mathematical exposition can have deŽnitions, lemmas, > propositions and theorems. The proof of a lemma or similar can cite > consequences from the statement of another but mustn¹t ever refer to > the proof - this would be as bad as code jumping into the middle of > a subroutine or the documentation of a method referring to the > code of another method. > I suggest an exposition of your work would have the same structure > with an executive summary at the top and cascading expansions of > each point. (A good book contains a series of chapters. Each chapter > starts out by saying what it is going to say, then says it in detail, > and Žnally ends by summarising what it has said) Your previous > website > made a point of being terse but this isn¹t necessary. I agree. Let¹s get started. I can make you rich, if you aren¹t rich already. If you are rich, I¹ll make you powerful. If you¹re already powerful, hell, why not just do it? > Other Viewpoints > It is not at all an easy thing to do, but if you could try to > explain, > in detail, other people¹s standpoints as well as your own then this > would be useful. I¹m thinking in particular of the factorisation > of certain polynomials in which you say that one root is coprime > to a prime factor of the constant term of the polynomial and other > people have shown explicit factorisations of the polynomial in > which this is not true. I didn¹t follow your rebuttal. > Dead Viewpoints > It would take great courage but if you could also document points > where you are no longer holding a former position then this also > would interesting. I¹m thinking in particular of whether a ring > must be closed under an inŽnite summation and whether Z[pi] is > isomorphic to the reals. > > *SimpliŽcations* > The polynomial that you use is very complicated with several > parameters. Is all the complication necessary to the proof and does > it add to it? Is the Œu¹ term necessary at all? You have resorted > to actually substituting values in order to explain - why not go > back and give the whole argument again but use a polynomial which > is as simple as possible. In fact, I don¹t always follow your > reasoning but perhaps it would be more obvious if you used a > quadratic polynomial instead of a trinomial or explained why the > argument doesn¹t work for quadratics. > *Style* > > I, too, like long sentences (my second choice for a book to take > if I were to be stranded on a desert island would be the collected > works of Jane Austen) and you are much better at writing them than > I am but they can be difŽcult for other people to follow. > Take this as an example: > . I already have as the polynomial is > . P(m) = f^2((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1+mf^2 )x u^2 + u^3f) > . and your attempts at confusing the issue by trying to push up x, > . don¹t help you here, as the g¹s as factors of m, necessarily have > . a constant term with respect to m, and your claim that it can vary > . with m, is nonsensical on its face. > It makes sense and I can understand it but I have to work at it. > I suggest that shorter sentences would make it easier for the > reader. > Penny Hassett I like it Penny Hassett, and I¹m willing to do some work, but not much as I¹m the engine that drive everything anyway. You prepare to do some work--and make no mistake you WILL work very hard--and I¹ll try to give you success. After all, I¹m already one of the most powerful men on the planet. With your help, I can get organized and maybe do some good in this world. Email me if you¹re interested, all offers are rescinded if you do not. James Harris === Subject: Re: Quick Math Guide to core error issues >>James, >>You post so much material that I don¹t have the time or desire to >>read everything and I lose track of your position and arguments. >>Some time ago you had an organised website with deŽnitions and cross- >>reference and you have posted several summaries and histories on this >>newsgroup which would be easier to refer to if they were permanently >>available. I, for one, would be pleased if you would resurrect your >>website and put the information back. > I like you Penny Hasset and appreciate your commentary which is why > I¹m posting in a thread where I don¹t need to post as it¹s a quide. > My problem is that I don¹t want to use MSN Groups, and I don¹t feel > like going to another website provider. > I *am* willing to allow someone else to host my work as long as they > give me complete editorial control. > Otherwise, it¹s easier for me to just post rather than try to maintain > a website. http:www.sphosting.com is easy to use, and allows Žle uploading. Of course, if you aren¹t willing to do any work, by all means keep everyone somewhat confused and off-balance. Perhaps you could post a weekly update? -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Quick Math Guide to core error issues > as I¹m the engine that drive everything anyway. > You prepare to do some work--and make no mistake you WILL work very > hard--and I¹ll try to give you success. > After all, I¹m already one of the most powerful men on the planet. > With your help, I can get organized and maybe do some good in this > world. > Email me if you¹re interested, all offers are rescinded if you do not. > James Harris of you material up on a web-site then I¹m willing to advise on the format and style by way of the sci.math newsgroup but that¹s all. PS. Lest you feel I am being dishonest when you Žnd out later, let me say that it will be obvious to many people in Britain that I am using a nom-de-keyboard. === Subject: Re: Quick Math Guide to core error issues > I like it Penny Hassett, and I¹m willing to do some work, but not much > as I¹m the engine that drive everything anyway. > > You prepare to do some work--and make no mistake you WILL work very > hard--and I¹ll try to give you success. > > After all, I¹m already one of the most powerful men on the planet. > > With your help, I can get organized and maybe do some good in this > world. > > Email me if you¹re interested, all offers are rescinded if you do not. > > James Harris > of you material up on a web-site then I¹m willing to advise on the > format and style by way of the sci.math newsgroup but that¹s all. last couple of days, partly out of EXTREME FRUSTRATION at my situation. I¹ve found that I can have fun with postings, which makes me feel better. Still I was sincere about the $250k but am now relieved that you declined. Oh yeah, I¹ve taken your advice though as I¹m using only m as a variable in my recent postings as I¹m *really* ready to Žnish things up. > PS. Lest you feel I am being dishonest when you Žnd out later, let > me say that it will be obvious to many people in Britain that > I am using a nom-de-keyboard. Oh hey, I¹d started calling you my money penny too. Kind of like a James thing, you know, Bond, James Bond. Oh well, maybe someday you¹ll have reason to give me your name, but it¹s not a big deal. In any event, unlike with Nora Baron, I won¹t put quotes around your name. James Harris === Subject: Re: Quick Math Guide to core error issues > I¹m willing to do some work, but not much I knew it! You¹re just too lazy! === Subject: Re: Quick Math Guide to core error issues putting aside the question of wether there is *any* real JSH, I still have to question this one¹s entitlement to the name, when he goes over the top and says that a correspondent can be proved to be in his newfound ring of what ever. back to teh question of the real one: if he¹s not being paid to do this, or has a pension that is allowing him to make fun of Whitey (?), then it really is pretty strange. note on Anglo-american history: slavery was a British institution; that¹s why they supported the Confederacy (along with the New York Times etc.) with ships & materiel, and actually organized the Civil War. (this makes for a good, revisionist question: What was our 3rd war with Great Britain?) [see http://tarpley.net] > I¹m posting in a thread where I don¹t need to post as it¹s a quide. > An explanation of what an algebraic integer is , some > examples and their general properties would be useful. > I vaguely recall that you had some software that would > take a monic quadratic equation with integer coefŽcients > and factorise it into linear terms with algebraic integer > coefŽcients. This could usefully go here as a Java applet. > You see, I¹ve found and dropped more mathematics than most people > Actually *you* are in it Penny Hasset, as the object ring is rather > large. > You see, you are a mathematical object which I can prove using some > rather basic logic and Goedel¹s proof. > I like it that you¹re in Britain. > I lose track of the arguments and how they are settled or even > if they are. Your executive summary below is a good start but that > is all it is. Hotlinks to expansions, with proofs, would be better. > If you¹re already powerful, hell, why not just do it? > It would take great courage but if you could also document points > where you are no longer holding a former position then this also > would interesting. I¹m thinking in particular of whether a ring > must be closed under an inŽnite summation and whether Z[pi] is > isomorphic to the reals. > . I already have as the polynomial is > . > . P(m) = f^2((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1+mf^2 )x u^2 + u^3f) > . > . and your attempts at confusing the issue by trying to push up x, > . don¹t help you here, as the g¹s as factors of m, necessarily have > . a constant term with respect to m, and your claim that it can vary > . with m, is nonsensical on its face. --UN HYDROGEN (sic; Methanex (TM) reformanteurs) ECONOMIE?... La Troi Phases d¹Exploitation de la Protocols des Grises de Kyoto: (FOSSILISATION [McCainanites?] (TM/sic))/ BORE/GUSH/NADIR @ http://www.tarpley.net/aobook.htm. Http://www.tarpley.net/bushb.htm (content partiale, below): 17 -- L¹ATTEMPTER de COUP D¹ETAT, 3/30/81 === Subject: Re: Quick Math Guide to core error issues > [...] > note on Anglo-american history: > slavery was a British institution; The British slave traders bought their slaves from Arab and African trade slavers, so it wasn¹t a British institution it was an international one. Slavery persists to this day of course. > [...] -- G.C. === Subject: Re: Quick Math Guide to core error issues > For those of you trying to keep up with the mathematical facts in the > discussions about the error in core mathematics from a problem with a > deŽnition, this post will outline the important ones quickly and > succinctly. > 1. First the problematic deŽnition: > Algebraic integers are deŽned to be roots of monic polynomials with > integer coefŽcient e.g. x^3 + 3x + 1 or x^234 - 34x^12 + 17, where > monic refers to the leading coefŽcient. > My assertion is that the over hundred year old deŽnition excludes > numbers that have to be included to keep from having contradiction > i.e. mathematical inconsistency. This is sheer idiocy. A deŽnition cannot lead to a contradiction. A deŽnition, that is not correctly understood by a wannabe maths genius like yourself, followed by some ridiculously confused attempts at proving things, can lead a sufŽciently stupid person to thinking there are contradictions. But that is _your_ problem, and not a problem of the deŽnition. === Subject: Re: Quick Math Guide to core error issues In sci.physics, James Harris discussions about the error in core mathematics from a problem with a > deŽnition, this post will outline the important ones quickly and > succinctly. > 1. First the problematic deŽnition: > Algebraic integers are deŽned to be roots of monic polynomials with > integer coefŽcient e.g. x^3 + 3x + 1 or x^234 - 34x^12 + 17, where > monic refers to the leading coefŽcient. > My assertion is that the over hundred year old deŽnition excludes > numbers that have to be included to keep from having contradiction > i.e. mathematical inconsistency. And these numbers are ... what? Presumably, you can produce a counterexample. > 2. The important tool I use is a polynomial: > P(m) = f^2((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1+mf^2 )x u^2 + u^3 f) > The form of the polynomial allows me to factor P(m) into > non-polynomial factors, and the factorization with those factors is > P(m) = (a_1 x + uf)(a_2 x + uf)(a_3 x + uf) > where the a¹s are roots of the following cubic: > a^3 + 3(-1+mf^2)a^2 - f^2(m^3 f^4 - 3m^2 f^2 + 3m). Pedant point: ITYM a^3 + 3(-1+mf^2)a^2 - f^2(m^3 f^4 - 3m^2 f^2 + 3m) = 0. > 3. Dispute centers around what happens when I divide P(m) by f^2, > which you¹ll note is a factor of the polynomial in the ring of > algebraic integers. That it is, for what it¹s worth. However, you¹ve not gotten around the f^(2/3) problem yet. I posit that a perfectly valid transformation of your cubic is Q(m) = P(m) / f^2 = (b_1 x + uf^(1/3))(b_2 x + uf^(1/3))(b_3 x + uf^(1/3)) where b_{i} = a_{i} / f^(2/3). In fact, that¹s probably what you¹d end up with anyway! :-) I, however, make no claims regarding the b_{i} being integers, algebraic or otherwise. I¹m not sure what can be deduced therefrom. It¹s worth noting that f^(1/3) is an algebraic integer if f is. Another transformation is R(m) = P(m) / f^3 = (c_1 x + u)(c_2 x + u) (c_3 x + u) although in this case we have a problem, as not all the c¹s are algebraic integers; there¹s a missing factor of 1/f in there somewhere. This factor can be added in: Q(m) = f R(m) = (c_1 x + u) (c_2 x + u) (c_3 fx + uf) but it could equally easily be added in: Q(m) = (c_1 f^(1/3) x + u f^(1/3)) (c_2 f^(1/3) x + u f^(1/3)) (c_3 f^(1/3)x + uf^(1/3)) or Q(m) = (c_1 x + u) ( c_2 f^(1/2) x + u f^(1/2) ) (c_3 f^(1/2) x + u f^(1/2)) I just don¹t know at this point. > 4. Mathematicians have argued that f^2 divides off as a function of m > because if they concede that it divides off independent of m, then I > can show that only two of the roots of > a^3 + 3(-1+mf^2)a^2 - f^2(m^3 f^4 - 3m^2 f^2 + 3m) > have f as a factor. If one sets f = 2, m = 1 we get a^3 + 9a^2 - 28 which has as one root a_1 = -2. Factoring, we get a^3 + 9a^2 - 28 = (a + 2) (a^2 + 7a - 14) so the other two roots are a_x = (-7 ± sqrt(49 + 56)) / 2 = -7/2 ± sqrt(105) / 2. Only one of these roots (-2) is divisible by 2. The other two roots -7/2 ± sqrt(105) / 2 are such that, if we set b_x = a_x/2, or a_x = 2b_x, we get b_x = -7/4 ± sqrt(105) / 4. What equation of integer coefŽcients does the b¹s satisfy? That¹s simple enough; substituting 2b for a, we get 4b^2 + 14b - 14 = 0 or 2b^2 + 7b - 7 = 0. Clearly, the b¹s are not algebraic integers, and therefore two of the original a¹s are not divisible by 2, for this particular setting of f and m. This is a counterexample to your original proposition. > 5. However, it turns out that if you go to the Želd of algebraic > numbers you can prove that for *certain* values of m and f, the roots > of the cubic do not have f as a factor *in the ring of algebraic > numbers* which is the inconsistency. > That is, for the math to be consistent, two of the roots *should* have > f as a factor as long as m and f are algebraic integers, but while I > can show they do for a particular values like m=1, f=sqrt(2), there > are other values you can show they do not *in the ring of algebraic > integers* which results from the deŽnition and its focus on monic > polynomials. I take it you want to include -7/4 ± sqrt(105) / 4 in the ring of algebraic integers? Please clarify. > Note: In the ring of algebraic integers you can¹t see the problem but > have to go to the Želd of algebraic numbers as from within the ring > of algebraic integers it appears that only two of the roots have a > factor that is f. 2/3 can be divided by 3 (the result being 2/9). Did you have a point here? > James Harris -- #191, ewill3@earthlink.net It¹s still legal to go .sigless. === Subject: Re: Quick Math Guide to core error issues > In sci.physics, James Harris > ... > 5. However, it turns out that if you go to the Želd of algebraic > numbers you can prove that for *certain* values of m and f, the roots > of the cubic do not have f as a factor *in the ring of algebraic > numbers* which is the inconsistency. > > That is, for the math to be consistent, two of the roots *should* have > f as a factor as long as m and f are algebraic integers, but while I > can show they do for a particular values like m=1, f=sqrt(2), there > are other values you can show they do not *in the ring of algebraic > integers* which results from the deŽnition and its focus on monic > polynomials. > I take it you want to include > -7/4 ± sqrt(105) / 4 > in the ring of algebraic integers? No, he can not add both. Their sum is -7/2, and adding both would make 2 a unit. He wants to add only one, but he will not tell us which one. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Quick Math Guide to core error issues In sci.physics, Dik T. Winter : > In sci.physics, James Harris > 5. However, it turns out that if you go to the Želd of algebraic > numbers you can prove that for *certain* values of m and f, the roots > of the cubic do not have f as a factor *in the ring of algebraic > numbers* which is the inconsistency. > > That is, for the math to be consistent, two of the roots *should* have > f as a factor as long as m and f are algebraic integers, but while I > can show they do for a particular values like m=1, f=sqrt(2), there > are other values you can show they do not *in the ring of algebraic > integers* which results from the deŽnition and its focus on monic > polynomials. > > I take it you want to include > -7/4 ± sqrt(105) / 4 > in the ring of algebraic integers? > No, he can not add both. Their sum is -7/2, and adding both would make > 2 a unit. He wants to add only one, but he will not tell us which one. It¹s a package deal. :-) And he gets 4 for the price of 2; the reciprocals need to be added as well, as unit * unit = unit and unit / unit = unit. In fact, a lot more will be dragged in by this inclusion. But the original deŽnition discriminates against this number (and for good reason). -- #191, ewill3@earthlink.net It¹s still legal to go .sigless. === Subject: Re: Quick Math Guide to core error issues > For those of you trying to keep up with the mathematical facts in the > discussions about the error in core mathematics from a problem with a > deŽnition, this post will outline the important ones quickly and > succinctly. > 1. First the problematic deŽnition: > Algebraic integers are deŽned to be roots of monic polynomials with > integer coefŽcient e.g. x^3 + 3x + 1 or x^234 - 34x^12 + 17, where > monic refers to the leading coefŽcient. > My assertion is that the over hundred year old deŽnition excludes > numbers that have to be included to keep from having contradiction > i.e. mathematical inconsistency. The deŽnition just deŽnes a set of numbers. The deŽnition itself cannot produce a contradiction. Contradictions are produced when one Œtheorem¹ contradicts another theorem. If you think you have a contradiction here, what is the known theorem in algebraic number theory which is being contradicted? (See below at {###] for my speculation on this.) > 2. The important tool I use is a polynomial: > P(m) = f^2((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1+mf^2 )x u^2 + u^3 f) > The form of the polynomial allows me to factor P(m) into > non-polynomial factors, and the factorization with those factors is > P(m) = (a_1 x + uf)(a_2 x + uf)(a_3 x + uf) > where the a¹s are roots of the following cubic: > a^3 + 3(-1+mf^2)a^2 - f^2(m^3 f^4 - 3m^2 f^2 + 3m). ... and the a¹s are therefore algebraic integers (this cubic is monic). > 3. Dispute centers around what happens when I divide P(m) by f^2, > which you¹ll note is a factor of the polynomial in the ring of > algebraic integers. > 4. Mathematicians have argued that f^2 divides off as a function of m > because if they concede that it divides off independent of m, then I > can show that only two of the roots of >[*] a^3 + 3(-1+mf^2)a^2 - f^2(m^3 f^4 - 3m^2 f^2 + 3m) > have f as a factor. No - we don¹t argue that it divides off as a function of m. We argue essentially that f^2 is distributed among the factors (ai*x + u*f) in a way which depends on m. For example, if m is such that the polynomial [*] that you give above is reducible, then one factor is relatively prime to f. But if m is such that [*] is irreducible, then *none* of the factors are relatively prime to f. In fact in the irreducible case, ALL of the factors (ai*x + u*f) are divisible by f^{2/3}. See my post of Oct 18 in the thread Finishing argument - core error proven for a proof of this. > 5. However, it turns out that if you go to the Želd of algebraic > numbers you can prove that for *certain* values of m and f, the roots > of the cubic do not have f as a factor *in the ring of algebraic > numbers* which is the inconsistency. The Žrst part is true: it happens whenever [*] is irreducible, which is true for most values of m. But it does not result in a contradiction. The factorization is different when the polynomial [*] is irreducible than when it is not. The twain do not meet (i.e., [*] is either irreducible or it is not) so there is no inconsistency. > That is, for the math to be consistent, two of the roots *should* have > f as a factor as long as m and f are algebraic integers, but while I > can show they do for a particular values like m=1, f=sqrt(2), there > are other values you can show they do not *in the ring of algebraic > integers* which results from the deŽnition and its focus on monic > polynomials. f = sqrt(2) is of no interest here. You original concern, relevant not only to your claims in Advanced Polynomial Factorization and Core error but also to your proof of Fermat¹s last theorem, dealt with f a a prime > 3 and m an integer relatively prime to f. Our counterexamples to your argument are restricted to the latter conditions. But even if one wanted to generalize for formal academic reasons: how f^2 distributes among the factors (a1*x + u*f) differs as described above for different combinations of m and f. Proving something about the form of the factorization for one combination does not prove it for others, as you inexplicably seem to believe. Similarly proving that for m = 0, f^2 distributes into the 3 factors as f, f, and 1, tells you nothing about cases for which m <> 0. I am surprised to see that you have mentioned your erroneous belief to the contrary, for the thousandth time. > Note: In the ring of algebraic integers you can¹t see the problem but > have to go to the Želd of algebraic numbers as from within the ring > of algebraic integers it appears that only two of the roots have a > factor that is f. A bizarre statement. In the Želd of algebraic numbers, every number has f as a factor! This is of no interest at all. The whole point of what you have been doing is lost if you decide to start talking about factorizations in a Želd. Nora B. > James Harris [###] Your result, if true, would contradict one of the following theorems: 1. Roots of non-monic primitive irreducible polynomial with integer coefŽcients cannot be algebraic integers. 2. The set of algebraic numbers constitutes a ring. I think you are refusing to say that your result contradicts 1. because you have gone on record (in 2002) as accepting that 1. is a correct theorem. You have not thought much about 2., which is also a theorem and one which is moderately difŽcult to prove. You have chosen to say that your result follows from an error in the deŽnition of algebraic integers because you know that would lead to the conclusion that mathematics is inconsistent (which you and the rest of us abhor), OR that your own proof is wrong. And your emotional state is such that you cannot possibly accept the latter conclusion. But saying that your result is a consequence of an erroneous deŽnition makes no sense at all on any scale. N.B. === Subject: Re: Quick Math Guide to core error issues Adjunct Assistant Professor at the University of Montana. [.snip.] >> 4. Mathematicians have argued that f^2 divides off as a function of m >> because if they concede that it divides off independent of m, then I >> can show that only two of the roots of >>[*] a^3 + 3(-1+mf^2)a^2 - f^2(m^3 f^4 - 3m^2 f^2 + 3m) >> have f as a factor. > No - we don¹t argue that it divides off as a function of >m. We argue essentially that f^2 is distributed among the >factors (ai*x + u*f) in a way which depends on m. For example, >if m is such that the polynomial [*] that you give above is >reducible, then one factor is relatively prime to f. This is not true in general either. If m=1 and f=2, one factor is a multiple of f, and the other two are multiples of proper factors of f; the polynomial [*] in that case factors as a product of a linear and an irreducible quadratic. What we have said is that: > But if >m is such that [*] is irreducible, then *none* of the factors >are relatively prime to f. But there have been no general conclusions about the reducible case in general, other than it may indeed be the case that one of the factors is coprime to f. Why do you take so much trouble to expose such a reasoner as Mr. Smith? I answer as a deceased friend of mine used to answer on like occasions - A man¹s capacity is no measure of his power to do mischief. Mr. Smith has untiring energy, which does something; self-evident honesty of conviction, which does more; and a long purse, which does most of all. He has made at least ten publications, full of Žgures few readers can criticize. A great many people are staggered to this extent, that they imagine there must be the indeŽnite something in the mysterious all this. They are brought to the point of suspicion that the mathematicians ought not to treat all this with such undisguised contempt, at least. -- A Budget of Paradoxes, Vol. 2 p. 129 by Augustus de Morgan Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Quick Math Guide to core error issues ... > 1. First the problematic deŽnition: > > Algebraic integers are deŽned to be roots of monic polynomials with > integer coefŽcient e.g. x^3 + 3x + 1 or x^234 - 34x^12 + 17, where > monic refers to the leading coefŽcient. > > My assertion is that the over hundred year old deŽnition excludes > numbers that have to be included to keep from having contradiction > i.e. mathematical inconsistency. > The deŽnition just deŽnes a set of numbers. The deŽnition > itself cannot produce a contradiction. Contradictions are > produced when one Œtheorem¹ contradicts another theorem. > If you think you have a contradiction here, what is the > known theorem in algebraic number theory which is being > contradicted? (See below at {###] for my speculation on this.) ... > Your result, if true, would contradict one of the > following theorems: > 1. Roots of non-monic primitive irreducible polynomial with > integer coefŽcients cannot be algebraic integers. > 2. The set of algebraic numbers constitutes a ring. You mean algebraic integers here. > I think you are refusing to say that your result contradicts > 1. because you have gone on record (in 2002) as accepting that > 1. is a correct theorem. You have not thought much about 2., > which is also a theorem and one which is moderately difŽcult to > prove. Actually he is also on record accepting that 2 is true, the last time was not so long ago. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Quick Math Guide to core error issues > For those of you trying to keep up with the mathematical facts in the > discussions about the error in core mathematics from a problem with a > deŽnition, this post will outline the important ones quickly and > succinctly. > 1. First the problematic deŽnition: > Algebraic integers are deŽned to be roots of monic polynomials with > integer coefŽcient e.g. x^3 + 3x + 1 or x^234 - 34x^12 + 17, where > monic refers to the leading coefŽcient. > My assertion is that the over hundred year old deŽnition excludes > numbers that have to be included to keep from having contradiction > i.e. mathematical inconsistency. > The deŽnition just deŽnes a set of numbers. The deŽnition > itself cannot produce a contradiction. Contradictions are > produced when one Œtheorem¹ contradicts another theorem. > If you think you have a contradiction here, what is the > known theorem in algebraic number theory which is being > contradicted? (See below at {###] for my speculation on this.) > 2. The important tool I use is a polynomial: > P(m) = f^2((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1+mf^2 )x u^2 + u^3 f) > The form of the polynomial allows me to factor P(m) into > non-polynomial factors, and the factorization with those factors is > P(m) = (a_1 x + uf)(a_2 x + uf)(a_3 x + uf) > where the a¹s are roots of the following cubic: > a^3 + 3(-1+mf^2)a^2 - f^2(m^3 f^4 - 3m^2 f^2 + 3m). > ... and the a¹s are therefore algebraic integers (this cubic > is monic). > 3. Dispute centers around what happens when I divide P(m) by f^2, > which you¹ll note is a factor of the polynomial in the ring of > algebraic integers. > 4. Mathematicians have argued that f^2 divides off as a function of m > because if they concede that it divides off independent of m, then I > can show that only two of the roots of >[*] a^3 + 3(-1+mf^2)a^2 - f^2(m^3 f^4 - 3m^2 f^2 + 3m) > have f as a factor. > No - we don¹t argue that it divides off as a function of > m. We argue essentially that f^2 is distributed among the > factors (ai*x + u*f) in a way which depends on m. For example, > if m is such that the polynomial [*] that you give above is > reducible, then one factor is relatively prime to f. But if > m is such that [*] is irreducible, then *none* of the factors > are relatively prime to f. In fact in the irreducible case, > ALL of the factors (ai*x + u*f) are divisible by f^{2/3}. > See my post of Oct 18 in the thread Finishing argument - core > error proven for a proof of this. > 5. However, it turns out that if you go to the Želd of algebraic > numbers you can prove that for *certain* values of m and f, the roots > of the cubic do not have f as a factor *in the ring of algebraic > numbers* which is the inconsistency. > The Žrst part is true: it happens whenever [*] is irreducible, > which is true for most values of m. But it does not result > in a contradiction. The factorization is different when the > polynomial [*] is irreducible than when it is not. The twain > do not meet (i.e., [*] is either irreducible or it is not) > so there is no inconsistency. > That is, for the math to be consistent, two of the roots *should* have > f as a factor as long as m and f are algebraic integers, but while I > can show they do for a particular values like m=1, f=sqrt(2), there > are other values you can show they do not *in the ring of algebraic > integers* which results from the deŽnition and its focus on monic > polynomials. > f = sqrt(2) is of no interest here. You original concern, relevant > not only to your claims in Advanced Polynomial Factorization and Core > error but also to your proof of Fermat¹s last theorem, dealt with f a > a prime > 3 and m an integer relatively prime to f. Our counterexamples > to your argument are restricted to the latter conditions. But even if one > wanted to generalize for formal academic reasons: how f^2 distributes among > the factors (a1*x + u*f) differs as described above for different > combinations of m and f. Proving something about the form of the > factorization for one combination does not prove it for others, as > you inexplicably seem to believe. > Similarly proving that for m = 0, f^2 distributes into the 3 > factors as f, f, and 1, tells you nothing about cases for which > m <> 0. I am surprised to see that you have mentioned your erroneous > belief to the contrary, for the thousandth time. > Note: In the ring of algebraic integers you can¹t see the problem but > have to go to the Želd of algebraic numbers as from within the ring > of algebraic integers it appears that only two of the roots have a > factor that is f. > A bizarre statement. In the Želd of algebraic numbers, every > number has f as a factor! This is of no interest at all. The > whole point of what you have been doing is lost if you decide to > start talking about factorizations in a Želd. > Nora B. > James Harris > [###] > Your result, if true, would contradict one of the > following theorems: > 1. Roots of non-monic primitive irreducible polynomial with > integer coefŽcients cannot be algebraic integers. > 2. The set of algebraic numbers constitutes a ring. > I think you are refusing to say that your result contradicts > 1. because you have gone on record (in 2002) as accepting that > 1. is a correct theorem. You have not thought much about 2., > which is also a theorem and one which is moderately difŽcult to > prove. You have chosen to say that your result follows from > an error in the deŽnition of algebraic integers because you > know that would lead to the conclusion that mathematics is > inconsiste